where \(\gamma =2|g_{k_{0}}|^{2}/v_{g}\)  is the spontaneous emission rate of the emitters to the waveguide with \(k_{0}=\omega _{0}/v_{g}\)  [1]}. \(\phi =k_{0}d\)  is the phase accumulation of photons traveling from one emitter to another through the waveguide and \(\Theta (t)\)  is the Heaviside step function. \(\kappa \)  denotes the loss of the emitters due to other decay channels, which can be much smaller than \(\gamma \)  experimentally.
